In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ith row contains only i elements.
Examples Notable particular examples include these:
The Bell triangle, whose numbers count the partitions of a set in which a given element is the largest singleton Catalan's triangle, which counts strings of matched parentheses Euler's triangle, which counts permutations with a given number of ascents Floyd's triangle, whose entries are all of the integers in order Hosoya's triangle, based on the Fibonacci numbers Lozanić's triangle, used in the mathematics of chemical compounds Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings Pascal's triangle, whose entries are the binomial coefficients Clark's triangle Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called generalized Pascal triangles; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.
Generalizations Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial. Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.
Applications Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers. The Boustrophedon transform uses a triangular array to transform one integer sequence into another. In general, a triangular array is used to store any table indexed by two natural numbers where j ≤ i.
Indexing Storing a triangular array in a computer requires a mapping from the two-dimensional coordinates (i, j) to a linear memory address. If two triangular arrays of equal size are to be stored (such as in LU decomposition), they can be combined into a standard rectangular array. If there is only one array, or it must be easily appended to, the array may be stored where row i begins at the ith triangular number Ti. Just like a rectangular array, one multiplication is required to find the start of the row, but this multiplication is of two variables (i*(i+1)/2), so some optimizations such as using a sequence of shifts and adds are not available.
See also Triangular number, the number of entries in such an array up to some particular row
References
External links Weisstein, Eric W., "Number Triangle", MathWorld
